| L(s) = 1 | + 2·3-s − 4·7-s + 9-s − 6·11-s − 2·13-s − 17-s + 4·19-s − 8·21-s − 5·25-s − 4·27-s − 4·31-s − 12·33-s + 4·37-s − 4·39-s + 6·41-s − 8·43-s + 9·49-s − 2·51-s + 6·53-s + 8·57-s + 4·61-s − 4·63-s − 8·67-s + 2·73-s − 10·75-s + 24·77-s + 8·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.51·7-s + 1/3·9-s − 1.80·11-s − 0.554·13-s − 0.242·17-s + 0.917·19-s − 1.74·21-s − 25-s − 0.769·27-s − 0.718·31-s − 2.08·33-s + 0.657·37-s − 0.640·39-s + 0.937·41-s − 1.21·43-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 1.05·57-s + 0.512·61-s − 0.503·63-s − 0.977·67-s + 0.234·73-s − 1.15·75-s + 2.73·77-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.628355001583189625988551593246, −8.657255224248218342106471033370, −7.76589645819579710746731573989, −7.25946473788827804639135121523, −6.03835056135927571761510178023, −5.19013856842488788481068619038, −3.78997975332835388697662085523, −2.95627382514208231650511213543, −2.34610746208050944602353534104, 0,
2.34610746208050944602353534104, 2.95627382514208231650511213543, 3.78997975332835388697662085523, 5.19013856842488788481068619038, 6.03835056135927571761510178023, 7.25946473788827804639135121523, 7.76589645819579710746731573989, 8.657255224248218342106471033370, 9.628355001583189625988551593246
